A splitting method for separable convex programming

Liu

Mathematical Problems in Engineering

2017

As a special three-block separable convex programming, the stable principal component pursuit (SPCP) arises in many different disciplines, such as statistical learning, signal processing, and web data ranking. In this paper, we propose a proximal fully parallel splitting method (PFPSM) for solving SPCP, in which the resulting subproblems all admit closed-form solutions and can be solved in distributed manners. Compared with other similar algorithms in the literature, PFPSM attaches a Glowinski relaxation factor ∈ ( √ 3/2, 2/ √ 3) to the updating formula for its Lagrange multiplier, which can be used to accelerate the convergence of the generated sequence. Under mild conditions, the global convergence of PFPSM is proved. Preliminary computational results show that the proposed algorithm works very well in practice.

Section: Numerical Resultsmentioning

confidence: 99%

A Proximal Fully Parallel Splitting Method for Stable Principal Component Pursuit

Liu

Mathematical Problems in Engineering

2017

“…Therefore, how to simplify or even remove the potentially expensive correction step is very important in improving the numerical performance of relevant splitting algorithms. Most recently, He et al [30] were so inspired to propose a novel splitting method in the combination of the Gauss-Seidel and the Jacobian decompositions, and extensive numerical results further verified the efficiency of the proposed method. In this paper, we study along this line and propose a new partially parallel splitting method, which follows the similar decomposition philosophy of [30], but with a much simplified iterative scheme.…”

Section: Introductionmentioning

confidence: 86%

“…Most recently, He et al [30] were so inspired to propose a novel splitting method in the combination of the Gauss-Seidel and the Jacobian decompositions, and extensive numerical results further verified the efficiency of the proposed method. In this paper, we study along this line and propose a new partially parallel splitting method, which follows the similar decomposition philosophy of [30], but with a much simplified iterative scheme. Moreover, our approach can be accelerated by a relaxation step, and, as a consequence, it is faster than the method in [30] (see Sect.…”

Section: Introductionmentioning

confidence: 86%

A partially parallel splitting method for multiple-block separable convex programming with applications to robust PCA

Hou

Yang

2015

Comput Optim Appl

We consider a multiple-block separable convex programming problem, where the objective function is the sum of m individual convex functions without overlapping variables, and the constraints are linear, aside from side constraints. Based on the combination of the classical Gauss-Seidel and the Jacobian decompositions of the augmented Lagrangian function, we propose a partially parallel splitting method, which differs from existing augmented Lagrangian based splitting methods in the sense that such an approach simplifies the iterative scheme significantly by removing the potentially expensive correction step. Furthermore, a relaxation step, whose computational cost is negligible, can be incorporated into the proposed method to improve its practical performance. Theoretically, we establish global convergence of the new method in the framework of proximal point algorithm and worst-case nonasymptotic O(1/t) convergence rate results in both ergodic and nonergodic senses, where t counts the iteration. The efficiency of the proposed method is further demonstrated through numerical results on robust PCA, i.e., factorizing from incomplete information of an

“…In recent years, some novel splitting methods tailored for iterative scheme (1.1) are well established in the context of prediction-correction method, and they can be accordingly grouped into two categories. The first one is Gauss-Seideltype splitting method, which uses a correction step to update the output (predictor) generated by iterative scheme (1.2), e.g., see [33][34][35]. The second one is Jacobiantype splitting method, which decomposes the quadratic term in a Jacobian way such that the resulting subproblems can be solved simultaneously.…”

Section: Augmented-lagrangian-based Splitting Methodsmentioning

confidence: 99%

A distributed Douglas-Rachford splitting method for multi-block convex minimization problems

Han

2015

Adv Comput Math

A customized Douglas-Rachford splitting method (DRSM) was recently proposed to solve two-block separable convex optimization problems with linear constraints and simple abstract constraints. The algorithm has advantage over the well-known alternating direction method of multipliers (ADMM), the dual application of DRSM to the two-block convex minimization problem, in the sense that the subproblems can have larger opportunity of possessing closed-form solutions since they are unconstrained. In this paper, we further study along this way by considering the primal application of DRSM for the general case m ≥ 3, i.e., we consider the multi-block separable convex minimization problem with linear constraints where the objective function is separable into m individual convex functions without coupled variables. The resulting method fully exploits the separable structure and enjoys decoupled subproblems which can be solved simultaneously. Both the exact and inexact versions of the new method are presented in a unified framework. Under mild conditions, we manage to prove the global convergence of the algorithm. Preliminary numerical experiments for extracting the background from corrupted surveillance video verify the encouraging efficiency of the new algorithm.Keywords Douglas-Rachford splitting method · Multi-block convex minimization problem · Alternating direction method of multipliers · Robust principal component analysis Mathematics Subject Classifications (2010) 90C25 · 65K10 · 94A08